如何计算满足约束的3×n网格用A、B、C三色填充的方式数

Alright, let's break down how to count the number of valid 3×n grids colored with A, B, C while meeting your two constraints. I'll walk through the reasoning step by step using combinatorics and the inclusion-exclusion principle—it's a clean, straightforward way to model this problem.

First, let's restate the constraints clearly to avoid confusion:

• Row constraint: No entire row can be a single color (each row must have at least two distinct colors across its n cells).
• Column constraint: No entire column can be a single color (each column must have at least two distinct colors across its 3 cells).

Step 1: Calculate total grids satisfying only the column constraint #

First, let's find how many grids exist where every column is valid (no column is monochromatic):

• For a single column: There are 3^3 = 27 total color combinations. Subtract the 3 monochromatic columns (all A, all B, all C), leaving 27 - 3 = 24 valid columns.
• For n columns: Since each column choice is independent, the total number of such grids is:

  S = 24^n
  

Step 2: Subtract grids that violate the row constraint (Inclusion-Exclusion Principle) #

Now we need to remove grids that satisfy the column constraint but have at least one monochromatic row. The inclusion-exclusion principle helps us avoid over-counting overlapping cases (like grids with two monochromatic rows).

Define the following terms:

• T1: Grids with valid columns AND row 1 is monochromatic.
• T2: Same as T1 but for row 2; T3 for row 3.
• T12: Grids with valid columns AND rows 1 + 2 are monochromatic.
• T13: Same as T12 but for rows 1 + 3; T23 for rows 2 + 3.
• T123: Grids with valid columns AND all three rows are monochromatic.

By inclusion-exclusion, the number of invalid grids (violating row constraint but satisfying column constraint) is:

T = T1 + T2 + T3 - T12 - T13 - T23 + T123

Let's calculate each term:

Calculating T1, T2, T3 #

Take T1 as an example:

• Choose the color for row 1: 3 options (A, B, C).
• For each column: Row 1 is fixed to this color, so the column can't be monochromatic (rows 2 and 3 can't both match row 1's color).
• Each column has 3*3 - 1 = 8 valid combinations (total 9 for rows 2+3, minus 1 monochromatic case).
• So T1 = 3 * 8^n, and T2 = T3 = T1. Adding them up:

  T1 + T2 + T3 = 9 * 8^n
  

Calculating T12, T13, T23 #

Take T12 as an example, split into two cases:

1. Rows 1 and 2 are the same color:
   • Choose the color: 3 options.
   • For each column: The column can't be monochromatic, so row 3 can't match the shared color of rows 1+2. This gives 2 options per column.
   • Contribution: 3 * 2^n
2. Rows 1 and 2 are different colors:
   • Choose colors for rows 1 and 2: 3*2 = 6 options (since they must differ).
   • For each column: Since rows 1 and 2 are different, the column can never be monochromatic—row 3 can be any of 3 colors.
   • Contribution: 6 * 3^n

• Total for T12: 3*2^n + 6*3^n. Since T13 and T23 are identical, adding them up:

  T12 + T13 + T23 = 3*(3*2^n + 6*3^n) = 9*2^n + 18*3^n
  

Calculating T123 #

This is grids where all three rows are monochromatic, and every column is valid:

• Each column is just the fixed color triplet of the three rows. For the column to be valid, this triplet can't be monochromatic.
• There are 3^3 - 3 = 24 valid color triplets (total 27 minus 3 monochromatic ones).
• Note: This value is constant regardless of n, since the entire grid repeats the same valid column n times. So:

  T123 = 24
  

Step 3: Final Formula #

Combine all terms to get the number of fully valid grids:

ans = 24^n - 9*8^n + 9*2^n + 18*3^n - 24

Verify with n=2 #

Let's plug in n=2 to confirm it matches your example:

• 24^2 = 576
• 9*8^2 = 9*64 = 576
• 9*2^2 = 9*4 = 36
• 18*3^2 = 18*9 = 162
• ans = 576 - 576 + 36 + 162 - 24 = 174

Perfect, that's exactly the example output you provided!

内容的提问来源于stack exchange，提问作者Love Babbar